Audit 4.9 · Audit: \(\mathrm{GL}_n(E)/\mathrm{GL}_n(F)\) satisfies (SG) over \(\mathbb F_q(\!(t)\!)\)

1. Labels. Audit 4.3, item 2.
2. Dominance of the lift. \(c=-w_0c\) means \(c_i=-c_{n+1-i}\). For \(n=2m\), \(c_m\ge c_{m+1}=-c_m\) gives \(c_m\ge0\); for \(n\) odd, \(c_{m+1}=0\le c_m\). So \(\lambda(c)\) is dominant, \(w_0\lambda=(0,\dots,0,c_m,\dots,c_1)\), and \(\lambda-w_0\lambda=(c_1,\dots,c_m,[0],-c_m,\dots,-c_1)=c\).
3. A point of type \(c\). \(w_0\in\mathrm{GL}_n(\mathbb F_q)\) has \(w_0\sigma(w_0)=w_0^2=1\), so \(w_0\in X(\mathcal O)=K\cdot1\) (Lang; \(L^+\mathrm{GL}_n\) is connected). With \(u\in K\), \(u\sigma(u)^{-1}=w_0\), and \(\sigma(t^\lambda)=t^\lambda\): \(t^\lambda u\cdot1=t^\lambda w_0t^{-\lambda}=w_0\,t^{w_0\lambda-\lambda}\), of Cartan type \(\operatorname{dom}(w_0\lambda-\lambda)=c\). This point equals \(t^\lambda\cdot w_0\in t^\lambda X(\mathcal O)\), so the coset \(t^\lambda K\) contributes \(1\).
4. Support. \(kt^\lambda\cdot y=kt^\lambda\,y\,t^{-\lambda}\sigma(k)^{-1}\in Kt^\lambda Kt^{-\lambda}K\subseteq\bigsqcup_{\nu\le\lambda-w_0\lambda}Kt^\nu K\), because \(Kt^{-\lambda}K=Kt^{-w_0\lambda}K\) with \(-w_0\lambda\) dominant. By Audit 4.3 the \(K\)-orbit is the Cartan type.
5. Generation. Proposition B of the classical page needs a partial order with finite lower sets on the \(K\)-orbits, the support bound, and a nonzero value on the top orbit; all three hold. Corollary C does not apply literally (\(t^\lambda\cdot1=1\)), as Proposition 2.10 notes.
6. Direct check, \(n=2\), \(q=3\) (labels \((c,-c)\), representatives \(\left(\begin{smallmatrix}0&t^c\\t^{-c}&0\end{smallmatrix}\right)\)):

$$T_{(1,0)}\ast\varphi_0=\mathbf 1_{(1,-1)}+4\,\mathbf 1_{(0,0)},\quad T_{(2,0)}\ast\varphi_0=\mathbf 1_{(2,-2)}+4\,\mathbf 1_{(1,-1)}+12\,\mathbf 1_{(0,0)},$$ $$T_{(3,0)}\ast\varphi_0=\mathbf 1_{(3,-3)}+4\,\mathbf 1_{(2,-2)}+12\,\mathbf 1_{(1,-1)}+36\,\mathbf 1_{(0,0)},\qquad T_{(1,1)}\ast\varphi_0=\varphi_0 .$$

The system is unitriangular, so every orbit indicator is generated.

7. \(n=1\) (Counterexample 2.3(3)): \(z\mapsto z/\bar z\) identifies \(X'\) with \(E^1\subseteq\mathcal O_E^\times\) (as \(v(Nz)=2v(z)\)), and Hilbert 90 for \(\mathcal O_E^\times\) gives \(E^1=\{u/\bar u:u\in\mathcal O_E^\times\}=K\cdot1\); \(C_K=\mathbb C\varphi_0\).
8. Consequence (correct as stated). The split, Galois and unitary \(\mathbb F_q\)-forms of \((\mathrm{GL}_n\times\mathrm{GL}_n,\mathrm{GL}_n)\) all have \(\mathcal C_k^{\mathrm{Sat}}=\mathcal C_k\) (Audit 4.2) and give (SG): holds, holds (Proposition 2.10), fails (Audit 4.5). So no property of \((G_k,X_k)\) alone is equivalent to (SG) over \(\mathbb F_q(\!(t)\!)\), and failure of (R) does not force failure of (SG).